nLnLuc5e: nLPar1/Par2 constructions
nLnLuc5e transforms a point (n1)LnPx into a similar point nLnPx by drawing lines through the n versions of (n1)LnPx in a reference nLine parallel to the omitted line.
• Suppose the reference nLine is called Ref.
• When drawing parallel lines through the n versions of (n1)LnPx the result will be an nLine called Par1.
• When drawing another generation (acting like Par1 is Ref) the resulting nLine will be called Par2.
• When the corresponding lines of Par1 and Par2 are parallel and Par1 and Par2 are perspective, then this Perspective Center will be called Homothetic Center Par1/Par2HC(nLnPx)).
Par1/Par2Levelup constructions on nLnPx
It appears that :
* nLnLuc5e(3LnPi) = 4LnPi for i = 1,...,11
* nLnLuc5e(nLnP8) = (n+1)LnP8 for n = 4,5,6,7,8,9, . . .
* nLnLuc5e(nLnP3) exists for n = 4,5,6,7 (then possible end of homothecy)
* nLnLuc5e(nLnP5) exists for n = 4,5,6 (then possible end of homothecy)
* nLnLuc5e(nLnP7) exists for n = 4,5,6,7,8 (9 gives calculation problems)
It appears that :
* nLnLuc5e(3LnPi) = 4LnPi for i = 1,...,11
* nLnLuc5e(nLnP8) = (n+1)LnP8 for n = 4,5,6,7,8,9, . . .
* nLnLuc5e(nLnP3) exists for n = 4,5,6,7 (then possible end of homothecy)
* nLnLuc5e(nLnP5) exists for n = 4,5,6 (then possible end of homothecy)
* nLnLuc5e(nLnP7) exists for n = 4,5,6,7,8 (9 gives calculation problems)
Examples
3Lpoint

4Lpoint

5Lpoint

6Lpoint


4LeP1
= QLP3

5LnP7. 5LoP1 (1:2)
(Par1/Par2HC(4LeP1))

None


4LnP12 =
4LnP3
= QLP4 = CC(H(2))

5LnP12 =
5LnP7. 5LnP3 (1:1)
(Par1/Par2HC(4LnP12))

6LnP12
No linear relation with known 6Lpoints.
(Par1/Par2HC(5LnP12))

Etc.


4LnP13
= QLP28 = CC(H(3))

5LnP13 =
No lin. rel. with known 5Lpoints.
(Par1/Par2HC(4LnP13))

6LnP13 =
No linear relation with known 6Lpoints.
(Par1/Par2HC(5LnP13))

Etc.


4LnP14 =
4Lnp2
= QLP29 = CC(H(2))

5LnP14 =
5LnP7. 5LnP5 (2:1)
(Par1/Par2HC(4LnP14))

6LnP14 =
No linear relation with known 6Lpoints.
(Par1/Par2HC(5LnP14))

Etc.

CC(H(i)) = Center of the 4LCentercircle wrt HofstadterPoint(i).
Par1/Par2 constructions on nLHofstadter Points
* In a 5Line starting with 4L nP13 (QLP28) as Central Point for the Component 4Lines it appears that 5LPar1 is homothetic with 5LPar2 giving a Homothetic Center 5LnP13.
* In a 6Line starting with 5L nP13 as Central Point for the Component 5Lines it appears that 6LPar1 is homothetic with 6LPar2 giving a Homothetic Center 6L nP13.
* In a 7Line starting with 6L nP13 as Central Point for the Component 5Lines it appears that 7LPar1 is homothetic with 7LPar2 giving a Homothetic Center 7L nP13.
* etc.
* In a 6Line starting with 5L nP13 as Central Point for the Component 5Lines it appears that 6LPar1 is homothetic with 6LPar2 giving a Homothetic Center 6L nP13.
* In a 7Line starting with 6L nP13 as Central Point for the Component 5Lines it appears that 7LPar1 is homothetic with 7LPar2 giving a Homothetic Center 7L nP13.
* etc.
This process can be repeated for all other known QLpoints generated from Hofstadter Points X(3), X(186), X(256), X(5961), X(5962), X(5963), X(5964).
Corresponding Central Points in the 4Line will be QLP4 (wrt X(3)), QLP28 (wrt X(186)), QLP29 (wrt X(256)).
Corresponding Central Points in the 4Line will be QLP4 (wrt X(3)), QLP28 (wrt X(186)), QLP29 (wrt X(256)).
I checked it graphically in Cabri for X(256) up to level n=6. Further drawings for n>6 were impossible because of the many internal calculations for the drawing software.
So I checked them with Mathematica Software.
Again there were limitations wrt the many internal calculations.
However there were no contra indications for:
X(3)
X(186) up to level n=8
X(256) up to level n=7
X(5961) up to level n=7
X(5962) up to level n=7
X(5963) up to level n=6
X(5964) up to level n=4
Therefore I feel confident enough for this conjecture :
Let 3LP(i) be nAngle Centers P(i) in a Triangle as described in QFGmessage #1872, where i <> 1, 0, 1.
Let 4LQ(i) be the Circumcenter of the 4 versions of 3LP(i) in a 4Line.
For these points nLPar1 will be homothetic with nLPar2 using (n1)LQ(i) as Central Point, producing new Homothetic Center nLQ(i), for all n > 4.